1. Conditional Probability – A Color Sequence Example

2. Probability CS Prints Out
Color Sequence
Probability CS Prints Out
BBBB
.10 = 10%
BGGB
.25 = 25%
RGGR
.05 = 05%
YYYY
.30 = 30%
BYRG
.15 = 15%
RYYB
Total
1.00 = 100%

3. Compute Pr{ blue shows 1st | blue shows 4th }

4. Probability CS Prints Out
Pr{ blue shows 1st | blue shows 4th }
Identify Prior Event
“Blue Shows 4th “
Color Sequence
Probability CS Prints Out
BBBB
.10 = 10%
BGGB
.25 = 25%
RYYB
.15 = 15%
Total
.50 = 50%

5. Pr{ blue shows 1st | blue shows 4th };
Compute Prior Probability
Pr{ Blue Shows 4th } =
Pr{ exactly one of BBBB, BGGB, RYYB shows } =
Pr{BBBB} + Pr{BGGB} + Pr{RYYB} =
= .50

6. Probability CS Prints Out
Pr{ blue shows 1st | blue shows 4th }
Identify Joint Probability
“Blue Shows 1st and 4th “
Color Sequence
Probability CS Prints Out
BBBB
.10 = 10%
BGGB
.25 = 25%
Total
.35 = 35%

7. Pr{ blue shows 1st | blue shows 4th }
Compute Joint Probability
Pr{ Blue Shows 1st and 4th } =
Pr{ exactly one of BBBB, BGGB shows } =
Pr{BBBB} + Pr{BGGB} =
= .35

8. Conditional Probability =
Joint Probability / Prior Probability
So, Pr{ Blue Shows 1st | Blue Shows 4th } =
Pr{ Blue Shows 1st and 4th }/Pr{Blue Shows 4th } =
.35/ .50= .70

9. Compute Pr{ green shows 2nd or 3rd | yellow shows }

10. Probability CS Prints Out
Pr{ green shows 2nd or 3rd | yellow shows };
Identify Prior Event
“Yellow Shows”
Color Sequence
Probability CS Prints Out
YYYY
.30 = 30%
BYRG
.15 = 15%
RYYB
Total
0.60 = 60%

11. Pr{ green shows 2nd or 3rd | yellow shows }
Compute Prior Probability
Pr{ Yellow Shows } =
Pr{ exactly one of YYYY, BYRG, RYYB shows } =
Pr{YYYY}+ Pr{BYRG}+ Pr{RYYB} =
= .60

12. Identify Joint Event
“Green Shows 2nd or 3rd and Yellow Shows”
No sequences meet this requirement, so
Pr{“Green Shows 2nd or 3rd and Yellow Shows”} = 0

13. Conditional Probability =
Joint Probability / Prior Probability So
Pr{ Green Shows 2nd or 3rd | Yellow shows } =
Pr{ Green Shows 2nd or 3rd and Yellow shows }/
Pr{Yellow shows } =
0 / .60= 0

14. Compute Pr{ yellow shows | red shows }

15. Pr{ yellow shows | red shows } Probability CS Prints Out
Identify Prior Event
“Red Shows”
Color Sequence
Probability CS Prints Out
RGGR
.05 = 05%
BYRG
.15 = 15%
RYYB
Total
0.35 = 35%

16. Compute Prior Probability
Pr{ Red Shows } =
Pr{ exactly one of RGGR, BYRG, RYYB shows } =
Pr{RGGR} + Pr{BYRG} + Pr{RYYB } =
= .35

17. Probability CS Prints Out
Pr{ yellow shows | red shows }
Identify Joint Probability
“ Yellow and Red Show “
Color Sequence
Probability CS Prints Out
BYRG
.15 = 15%
RYYB
Total
0.30 = 30%

18. Compute Joint Probability
Pr{ Yellow and Red Show } =
Pr{ exactly one of BYRG, RYYB shows } =
Pr{BYRG}+ Pr{RYYB } =
= .30

19. Conditional Probability =
Joint Probability / Prior Probability So
Pr{ Yellow Shows | Red Shows } =
Pr{ Yellow and Red Shows }/Pr{Red Shows } =
.30/.35 =
6/7 =
.8571